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Mathematics of Computation

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Rapid multiplication modulo the sum and difference of highly composite numbers

Author: Colin Percival
Journal: Math. Comp. 72 (2003), 387-395
MSC (2000): Primary 65G50, 65T50; Secondary 11A51
Published electronically: March 5, 2002
MathSciNet review: 1933827
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Abstract: We extend the work of Richard Crandall et al. to demonstrate how the Discrete Weighted Transform (DWT) can be applied to speed up multiplication modulo any number of the form $a \pm b$ where $\prod _{p|ab}{p}$ is small. In particular this allows rapid computation modulo numbers of the form $k \cdot 2^n \pm 1$. In addition, we prove tight bounds on the rounding errors which naturally occur in floating-point implementations of FFT and DWT multiplications. This makes it possible for FFT multiplications to be used in situations where correctness is essential, for example in computer algebra packages.

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Additional Information

Colin Percival
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada

Keywords: Rapid multiplication, FFT, rounding errors
Received by editor(s): September 12, 2000
Received by editor(s) in revised form: March 15, 2001
Published electronically: March 5, 2002
Additional Notes: This work was supported by MITACS and NSERC of Canada
Article copyright: © Copyright 2002 American Mathematical Society